What are sine, cosine and tangent?

Labeling the three sides of a right triangle

To find sine, cosine, and tangent, you must first label the triangle's sides relative to one chosen acute angle. The hypotenuse is always the longest side, directly across from the 90-degree right angle. Its position never shifts.

The other two sides depend entirely on which acute angle you pick. The side directly facing your chosen angle across the triangle is the opposite leg. The side touching your angle, running between that angle and the right angle, is the adjacent leg.

The three trigonometric ratios

Sine (written as sin) compares the opposite leg to the hypotenuse: sin = opposite / hypotenuse. Because the hypotenuse is always the longest side, the sine of an acute angle is always between 0 and 1.

Cosine (written as cos) compares the adjacent leg to the hypotenuse: cos = adjacent / hypotenuse. Like sine, cosine always falls between 0 and 1 for an acute angle. Tangent (written as tan) compares the two legs directly: tan = opposite / adjacent. Because either leg can be longer, tangent can be any positive value.

These ratios are pure numbers without units. When you divide 3 inches by 5 inches, the units cancel out, leaving a pure ratio of 0.6.

A step-by-step example with numbers

Picture a right triangle with legs measuring 3 and 4, and a hypotenuse measuring 5. Choose the angle facing the side of length 3.

First, identify your sides: the opposite side is 3, the adjacent side is 4, and the hypotenuse is 5. Next, write the ratios: sine is 3/5 (or 0.6), cosine is 4/5 (or 0.8), and tangent is 3/4 (or 0.75).

Notice that dividing sine by cosine gives 0.6 / 0.8 = 0.75, which matches the tangent exactly. Tangent always equals sine divided by cosine.